= Mellin continuation of a nonprincipal Dirichlet L-function
{c}
For a <nonprincipal Dirichlet character> modulo $N$, set
$$
H_\chi(t)=\frac{\sum_{a=1}^N\chi(a)e^{-at}}{1-e^{-Nt}}.
$$
The numerator vanishes at $t=0$, so $H_\chi$ is analytic there and exponentially decreasing as $t\to\infty$. Initially for $\operatorname{Re}s>1$, $\Gamma(s)L(\chi,s)=\int_0^\infty H_\chi(t)t^{s-1}\,dt$. This integral already extends holomorphically to $\operatorname{Re}s>0$. Subtracting finite <Taylor polynomials> near zero, as in <meromorphic continuation of a Mellin transform from an asymptotic expansion>, continues it with possible simple poles only at nonpositive integers. Zeros of the reciprocal <Gamma function> cancel these, proving that $L(\chi,s)$ is an <entire function>. This does not require a <primitive Dirichlet character>.
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